Bài 51 trang 216 Đại số 10 Nâng cao: Chứng minh rằng nếu ∝ + β + γ = π thì...

Chứng minh rằng nếu (∝ + β + γ = π) thì

a) (sin alpha  + sin eta  + sin gamma  = 4cos {alpha  over 2}cos {eta  over 2}cos {gamma  over 2})

b) (cos alpha  + cos eta  + cos gamma  = 1 + 4sin {alpha  over 2}sin {eta  over 2}sin {gamma  over 2})

c) (sin2∝ + sin2β + sin2γ = 4sin∝ sinβ sin γ)

d) (co{s^2} propto + { m{ }}co{s^2}eta + co{s^2}gamma { m{ }}= 1 – 2cos∝ cosβ cosγ)

 Đáp án

a) Ta có:

(eqalign{
& sin alpha + sin eta + sin gammacr& = sin alpha + 2sin {{eta + gamma } over 2}cos {{eta – gamma } over 2} cr
& = sin alpha + 2sin {{pi – alpha } over 2}cos {{eta – gamma } over 2} cr&= 2sin {alpha over 2}cos {alpha over 2} + 2cos {alpha over 2}  cos {{eta – gamma } over 2} cr
& = 2cos {alpha over 2}(sin {alpha over 2} + cos {{eta – gamma } over 2})cr& = 2cos {alpha over 2}{ m{[sin}}{{pi – (eta + gamma )} over 2} + cos{{eta – gamma } over 2}{ m{]}} cr
& = 2cos {alpha over 2}(cos{{eta + gamma } over 2} + cos {{eta – gamma } over 2}) cr
& =4cos {alpha over 2}cos {eta over 2}cos {gamma over 2} cr} )

b) Ta có:

(eqalign{
& cos alpha + cos eta + cos gamma cr&= 2cos {{alpha + eta } over 2}cos {{alpha – eta } over 2} + 1 – 2sin {{2gamma } over 2} cr
& = 2cos ({pi over 2} – {gamma over 2})cos{{alpha – eta } over 2} + 1 – 2{sin ^2}{gamma over 2} cr&= 1 + 2sin {gamma over 2}(cos{{alpha – eta } over 2} – sin {gamma over 2}) cr
& = 1 + 2sin {gamma over 2}(cos{{alpha – eta } over 2} – cos{{alpha + eta } over 2}) cr
& = 1 + 4sin {alpha over 2}sin {eta over 2}sin {gamma over 2} cr} )

c) (sin2∝ + sin2β + sin2γ)

(= 2sin (∝ + β)cos(∝ – β ) + 2sinγcosγ)

(= 2sinγ (cos(∝ – β ) – cos(∝ + β)) )

(= 4sin∝ sinβ sin γ)

d) Ta có:

(eqalign{
& co{s^2} propto + { m{ }}co{s^2}eta + co{s^2}gamma { m{ }} cr
& { m{ = }}{{1 + cos 2alpha } over 2} + {{1cos 2eta } over 2} + {cos ^2}gamma cr
& = 1 + {1 over 2}(cos2alpha + cos 2eta ) + {cos ^2}gamma cr
& = 1 + cos (alpha + eta )cos(alpha – eta ) + {cos ^2}gamma cr
& = 1 + cos gamma (cos gamma – cos (alpha – eta )) cr&= 1 – cos gamma { m{[cos(}}alpha { m{ + }}eta { m{) + cos(}}alpha { m{ – }}eta ){ m{]}} cr
& = { m{ }}1{ m{ }}-{ m{ }}2cos propto { m{ }}coseta { m{ }}cosgamma cr} )